Coherent X-Ray Cherenkov Radiation (CXCR)
Produced by Microbunched Beams
M.A. Aginian1, K.A. Ispirian1, M.K. Ispiryan1, M.I. Ivanian2
1Alikhanian National Laboratory, Yerevan Physics Institute,
Br. Alikhanians 2, Yerevan, 0036, Armenia
2 CANDLE Synchrotron Research Institute, Acharyan 31, 0040
Yerevan, Armenia
RREPS-MEGHRI 2013
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1. Introduction
The density distribution of the Gaussian e-beams of FELs is
microbunched according to
N b exp( r 2 / 22r ) exp(  z 2 / 22z )
f ( r, z ) 
[1  b1 cos( kr , z )]
2
1/ 2
2r
( 2 )  z
,
with form factor
(1)
2
2
2 
2 
   2  2 
 
 




2
z
z
  b1 exp     k r  z 
  b1 exp     k r 
F ( , )  exp  ( k r ) exp 
2
 V
 V
 2 
 2 
  2V 


2
(2)
where  r  2 / kr  2c / r  Lund (1  K 2 / 2) / 2  2 and b1 are the
b
microbunching wavelength and the modulation depth, Nb, Lund and
K…
The coherent radiation of microbunched beams has been studied:
in [1]-COTR; in [2]-CXTR; in [3]- CXDR; in [4]- CChR; in [5]CPXR in [6]-CCUR; in [7]-a review of all.
1. J. Rosenzweig, G. Travish, A. Tremaine, Nucl. Instr. and Meth. A 365 (1995) 255.
2. E.D. Gazazian, K.A. Ispirian, R.K. Ispirian, M.I. Ivanian, Nucl. Instr. and Meth. B173 (2001) 160.
3. M.A. Aginian, K.A. Ispirian, M. Ispiryan, Proc. of NAS of Armenia, Physics, 47 (2012) 83.
4. M.A. Aginian, L.A. Gevorgian, K.A. Ispirian, to be publ. in Proc. of Intern. Conf. Channeling 2012.
5. M.A. Aginian, L.A. Gevorgian, K.A. Ispirian, to be publ. in Proc. of Intern. Conf. Channeling 22012.
6. L.A. Gevorgian, K.A. Ispirian, A. Shamamian, Nucl. Instr. and Meth. B 309 (2013) 63.
Theoretically substituting the calculated values of χ’ and χ” into
X-ray transition radiation (XTR) formulae (see [8]) and
experimentally using E=1.5 GeV [9] and 5-10 MeV [10-12] electrons
it has been shown that at K-. L- edges in narrow regions where
n(ω)>1, X-ray Cherenkov radiation (XCR) is produced.
This work is devoted to the theoretical study of coherent X-ray
Cherenkov radiation (CXCR) in water window region (ћω=285550 eV) which can be produced and used at DESY FLASH2 with
E=1.2 GeV [13] and at Japan SACLA [14] and DESY EuroFEL [15]
with E=8 and 10 GeV electrons.
8. G.M. Garibian, Yan Shi, Rentgenovskoe Perekhodnoe Izluchenie, Publishing home of Academy of
Sciences of Armenia, Yerevan, Armenia, 1983.
9. V.A. Bazilev et al, Pisma c Zh. Eksp. Teor. Fiz. 24 (1976) 406; JETF, 81 (1981) 1664.
10. W. Knulst et al, Appl. a) Phys. Lett. 79 (2001) 2999; b) 83 (2003) 4050; c)Proc. SPIE, 5196 (2004) 393.
d) W. Knults, Dissertation, Eindhoven University of Technology, 2004.
11. V. A. Bazilev, N.K. Zhevago, Izluchenie Bistrikh Chastits v Veshchestve i vo Vneshnikh
Polyakh, Moscow, Nauka, 1987.
12. K.A. Ispirian, Lecture “X-Ray Cherenkov Radiation”, in Proc. of NATO ARW 2004, Vol. 199, p. 217.
13. a) V. Ayvazyan et al, Eur. Phys. J. D37 (2006) 217; b) K. Honkavaara et al, Proc. of IPAC2012,
TUPP052, 1715, 2012; c) M. Vogt et al, Proc. IPAC 2013, TUPEAO04.
14. T. Ishikawa et al, Nature Photonics, 6, 540, 2012.
3
15. TESLA Technical Design Report 2001 DESY 011.
2. The Calculation of CXCR
Using the well known relation (see [16])
d 2 N CXCR
d 2 N XCR
2
 N b F (, )
dd
dd ,
Where the form-factor is given by (2) and
2
(3)
d N XCR
dd
by the XTR formula (1.59) of [8] which can be used also for XCR
d 2 N XTR d 2 N XCR 2 3
(  ' 2   ' '2 )


. (4)
2
2 2
2
2
2
2
dd dd
 (   ) [(     ' )   ' ' ]
Substituting (4) and (2) into (3) one derives for the spectral-angular
distribution of CXCR
2
3
 

d 2 N CXCR
('2 ' '2 )

2 2 2
2
2

 N b b1
exp  (kr ) exp    kr   z
 V

dd
 (2   2 )2 [( 2   2  ' )2  ' '2 ]



16. N. A. Korkhmazian, L.A. Gevorgian, M. L.Petrosian, Zh. Techn. Fiz. 47, 1583, 1977.
4
.(5)
The angular distribution of CXCR photon number is
obtained integrating (5) over ω
(' ' ' ) |
.
dN
2 V
N b
exp  (k  ) 
2
3
2 2
b 1
CXCR
d
2 2
2
r
2
2
 z r (   ) [(     ' |r )  ' ' |r ]
2
2
2
r
2
(6)
r
The spectral distribution of CXCR is obtained by integration
of (5) over θ
2
2
2
 

dN CXCR

2 2 2( '  ' ' )
 N b b1
I , 0 exp     kr  2z 
 V

d




where
,
(7)
0
3
2


I , 0    2
exp

(
k


)
d
r
2 2
2
2
2
2
(   ) [(     ' )  ' ' ]
0
, (8)
The upper limit of integration in (8) must be taken 0  1 /  since
the radiation intensities decrease sharply after these angles
For   1 one obtains
j

I  ,   
A  A*   B
where
2  
     
1e
0,  k     j 
1 1  C  2 ln k  
,
(9)
A
, B
 k
2
2
r
  j 
2
   j 
r
2

a, z    t a 1e t dt
z
    
2
r
2
, C=0.577216 (10)
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The total number of CXCR photons per bunch obtained by
integration of (6) over θ, or of (7) over ω is equal to
NCXCR  Nb2b12
2V (  '2   ' '2 ) | r
 z  r
I r , 0  .
(11)
3. Numerical Results
Obtained with the help of (6) - (11) for C are calculated for ћωr=
284 eV (FLASH2) with E=1.2 GeV [13], and SACLA [14] and
EuroFEL [15] with E=8 and 10 GeV, respectively, by “tuning”
the parameters. The results for Ti are calculated for SACLA [14]
and EuroFEL [15]at E=8 and 10 GeV, respectively.In all the cases
it is taken b1=1,  r  13m
 z  15m
.
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3a. The Angular Distributions of CXCR
a)
b)
Fig.1 a) and b) for C and Ti, respectively.
It is seen that θ are less than 1/γ.
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3b. The Spectral Distributions of CXCR
a)
b)
Fig.2 The Spectral Distributions of CXCR for C and Ti, a) and b),
respectively
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3c. The Energy Dependence of CXCR
a) Fig.2 a) and b) for C and Ti b)
It is seen that N increases slightly more weakly than ~ γ4.
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4. Discussion and Conclusions
For comparison Table gives the main parameters of SASE, XCR
and CXCR radiation for C expected at FLASH at E=1.2 GeV.
SASE
1012
XCR
3x106
CXCR
1.8x109
N
Δω/ω (%)
1.0
0.35
0.0014
Δω (eV)
2.8
1.0
0.027
The situation becomes much better for higher energy electrons and
Ti radiator. However, it has been taken b1=1, and any real b1<1
reduces the CXCR intensity ~b12 . In our knowledge, the values of b1
have not been measured anywhere.
Let us note that as for the first real X-ray laser [17] the CXCR beam
has more stable ћω line since it is the characteristics of radiator atoms
17. N.Rohringer et al, Nature, 481 (2012) 488.
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Summarizing one can make the following conclusions:
1)New type of radiation, CXCR, has been predicted and studied.
2)The use of microbunched beams, which are now sent to dumps,
can provide CXCR beams very monochromatic and stable. As the
beams of [17] they can find application in atomic physics and
for the measurement of b1. Indeed, if by another simple method, say,
measuring the current, one knows the value of Nb, then the
measured intensity can serve for determination of b1.
Thank you
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